Is the closed string an unknot?

[Master replies:]

A unknot is pretty much a circle or a 2-sphere, so is a closed string, maybe a bit more irregular but approximatly the same. Of course more is needed to identify the closed string as an unknot than the shape.
I assume that it is not so, yet I have no reason to deny it. I hope you may give me a reason.
Knots can also be added simply connecting the two knots. Is such a addition also possible with closed strings, if they are in fact unknots? Can, if it is a unknot, the closed string me knoted further?

 

[andrewkirk]

An unknot is a simple closed curve in 3D Euclidean space that is homotopic to a circle. Ie there must be a continuous deformation that maps it to, say the curve
$f:[0,1]\to\mathbb{R}^3:\ \ f(t)=(1,t,0)$ where the latter representation is in spherical coordinates. Defining a knot is easy. The hard bit is working out whether any particular simple closed curve in $\mathbb{R}^3$ is a knot.

It's not clear what your question is. You ask whether an unknot can be knotted 'further'. That depends what operations you allow. If you allow cutting and splicing then yes - just cut it, do a granny knot, then splice the ends together. If not, and the 'knotting' transformation must be continuous, then the answer is no, because whatever transformation you do to try to knot it can be reversed to transform it back into the unit circle.

 

[Hornbein]

A unknot is pretty much a circle or a 2-sphere, so is a closed string, maybe a bit more irregular but approximatly the same. Of course more is needed to identify the closed string as an unknot than the shape.
I assume that it is not so, yet I have no reason to deny it. I hope you may give me a reason.
Knots can also be added simply connecting the two knots. Is such a addition also possible with closed strings, if they are in fact unknots? Can, if it is a unknot, the closed string me knoted further?

To a physicist or mathematician, a knots is defined as a closed curve. All closed curves are knots, all knots are closed curves. This is NOT the definition used in standard English. It is quite different from the "knots" encountered in everyday life, which we won't discuss here.

The only way to change a knot to a topologically different knot is to cut the string, move it around, then paste it back together. This is a definition of "topologically different."

The circle is called the "unknot" because in physics often it can dissipate by shrinking to a point. No other knot can do that. So the unknot is often less stable than the other knots.

 

[Master replies:]

It's not clear what your question is. You ask whether an unknot can be knotted 'further'. That depends what operations you allow. If you allow cutting and splicing then yes - just cut it, do a granny knot, then splice the ends together. If not, and the 'knotting' transformation must be continuous, then the answer is no, because whatever transformation you do to try to knot it can be reversed to transform it back into the unit circle.

My question was not if the unknot may be knoted further, I apologize if that was unclear, but rather if the unknot is a primitve form of a higher more complex knot. I do not imagine so but...
But more importantly if the closed string may be considered as one and if it is any help to mathematics or the physics and if one may add two closed strings as one may do with knots.

 

[andrewkirk]

My question was not if the unknot may be knoted further, I apologize if that was unclear, but rather if the unknot is a primitve form of a higher more complex knot.

No, it isn't.

But more importantly if the closed string may be considered as one and if it is any help to mathematics or the physics and if one may add two closed strings as one may do with knots.

I'm afraid I do not understand what you are trying to ask here.